![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > ofcc | Structured version Visualization version GIF version |
Description: Left operation by a constant on a mixed operation with a constant. (Contributed by Thierry Arnoux, 31-Jan-2017.) |
Ref | Expression |
---|---|
ofcc.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
ofcc.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
ofcc.3 | ⊢ (𝜑 → 𝐶 ∈ 𝑋) |
Ref | Expression |
---|---|
ofcc | ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f/c 𝑅𝐶) = (𝐴 × {(𝐵𝑅𝐶)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ofcc.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
2 | fnconstg 6797 | . . . 4 ⊢ (𝐵 ∈ 𝑊 → (𝐴 × {𝐵}) Fn 𝐴) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 × {𝐵}) Fn 𝐴) |
4 | ofcc.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
5 | ofcc.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑋) | |
6 | fvconst2g 7222 | . . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑥) = 𝐵) | |
7 | 1, 6 | sylan 580 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑥) = 𝐵) |
8 | 3, 4, 5, 7 | ofcfval 34079 | . 2 ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f/c 𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶))) |
9 | fconstmpt 5751 | . 2 ⊢ (𝐴 × {(𝐵𝑅𝐶)}) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)) | |
10 | 8, 9 | eqtr4di 2793 | 1 ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f/c 𝑅𝐶) = (𝐴 × {(𝐵𝑅𝐶)})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 {csn 4631 ↦ cmpt 5231 × cxp 5687 Fn wfn 6558 ‘cfv 6563 (class class class)co 7431 ∘f/c cofc 34076 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-ofc 34077 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |